Page:Elektrische und Optische Erscheinungen (Lorentz) 136.jpg

If we denote (for the incident, reflected and transmitted light) the values of the magnitude just mentioned in the immediate vicinity of the marginal surface by

$\varphi _{1}(t),\varphi '_{1}(t)$ and $\varphi _{2}(t)$,

and the values of m by

$m_{1},m'_{1}$ und $m_{2}$

then we obtain as marginal conditions

$\varphi _{1}(t)+\varphi '_{1}(t)=\varphi _{2}(t)$

and

$m_{1}{\frac {\partial \varphi _{1}(t)}{\partial t}}+m'_{1}{\frac {\partial \varphi '_{1}(t)}{\partial t}}=m_{2}{\frac {\partial \varphi _{2}(t)}{\partial t}}$.

The last formula leads — when we neglect additive constants — to

$m_{1}\varphi _{1}(t)+m'_{1}\varphi '_{1}(t)=m_{2}\varphi _{2}(t)$,

and it is further given by elimination of $\varphi _{2}(t)$

$\varphi '_{1}(t)={\frac {m_{1}-m_{2}}{m_{2}-m'_{1}}}\varphi _{1}(t)$.

Now, that the amplitude of the reflected beam (at constant direction of the incident light) depends on the refractive index of the second body, stems from the fact, that, as it can easily be seen, $m_{2}$ changes with this exponent.

Now, in the next paragraph it should be demonstrated, that this $m_{2}$ (as long as the direction of the incident relative ray remains the same) is not affected by a translation in the direction of the z-axis. If it would be allowed to assume, that also with respect to a moving plate, the marginal conditions consist of the continuity of a certain magnitude $\varphi$ and its derivatives, then, at least for light polarized in the plane of incidence, we would have demonstrated the impossibility of the phenomenon sought by Fizeau. However, in reality the assumption on the marginal conditions is not allowed without closer investigation; the things said show at least, however, that the moving plate in no ways acts as a stationary one of somewhat different refractive index.